Consider the polar equation r= a cos(8) + b sin(8). (a) Express the equation in rectangular coordinates, and use this to show that the graph of the equation is a circle. What are the center and radius? r = a cos(8) + b sin(6) 2 = ar cos(6) + x2 + y2 = ax + by - ax + - by - 0 x2 - ax +? + y2 - by +62 = 2 + ( + (v - )' - ? + 63) Thus, in rectangular coordinates the center is (x, y) = and the radius is

Consider the polar equation r= a cos(8) + b sin(8). (a) Express the equation in rectangular coordinates, and use this to show that the graph of the equation is a circle. What are the center and radius? r = a cos(8) + b sin(6) 2 = ar cos(6) + x2 + y2 = ax + by - ax + - by - 0 x2 - ax +? + y2 - by +62 = 2 + ( + (v - )' - ? + 63) Thus, in rectangular coordinates the center is (x, y) = and the radius is

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter8: Complex Numbers And Polarcoordinates

Section: Chapter Questions

Problem 2GP

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Question

Consider the polar equation r= a cos(8) + b sin(8).
(a) Express the equation in rectangular coordinates, and use this to show that the graph of the equation is a circle. What are the center and radius?
r = a cos(8) + b sin(6)
2 = ar cos(0) +
x2 + y2 = ax + by
2 - ax +(
- by = 0
2 - by + 5? =
(x-) + (v - ±)° - ? + 63)
x2 -
- ax +
Thus, in rectangular coordinates the center is (x, y) =
and the radius is

Expert Solution

Step 1

Given:

The polar equation,

$r = a \cos (θ) + b \sin (θ)$

To find:

Center and radius.

Step 2

Consider the polar equation,

$r = a \cos (θ) + b \sin (θ) . . . (1)$

Converting the cartesian coordinates,

$x = r \cos θ, y = r \sin θ$

$\Rightarrow \cos θ = \frac{x}{r}, \sin θ = \frac{y}{r}$

The polar equation becomes,

$r = a \cdot \frac{x}{r} + b \cdot \frac{y}{r} r = \frac{a x}{r} + \frac{b y}{r}$

$\Rightarrow r^{2} = a x + b y$

But $r^{2} = x^{2} + y^{2}$

Step by step

Solved in 3 steps

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